Collateral choice and the fundamental theorem of asset pricing

In the classical quantitative finance literature it is assumed that there is a risk free rate at which hedgers can borrow and lend in the dynamic replication process of financial derivatives. In such a framework, under complete market conditions and absence of arbitrage opportunities, for a given numeraire whose price cannot vanish, prices of self financing portfolios divided by the numeraire behave like a martingales under a unique martingale measure associated with the numeraire. Nevertheless, in the current market environment a high percentage of deals are collateralized due to counperparty credit risk concerns. Depending on the collateral agreement, collateral can be in the form of cash in different currencies, but also in the form of assets (bonds, shares,...). In this paper we explore how the fundamental valuation theorem and the change of numeraire tollkit is reformulated under this new framework.